Angel "Java" Lopez en Blog

Publicado el 8 de Diciembre, 2014, 6:01

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Comencemos leyendo los párrafos iniciales del "paper" de Heisenberg. He conseguido una traducción al inglés (el original fue publicado en alemán) en el libro B. L. van der Waerden, editor, Sources of Quantum Mechanics (Dover Publications). Leo ahí el "paper 12":

Quantum-Theoretical Re-Interpretation of Kinematic and Mechanical Relations
W.Heisenberg

Vemos que Heisenberg menciona cinemática, porque se va a ocupar no sólo de la dinámica, sino también cómo podemos manejar conceptos como la posición x.

The present paper seeks to establish a basis for theoretical quantum mechanics founded exclusively upon relationships between quantities which in principle are observable.

Eso de "observable" es discutible. Pero Heisenberg está motivado por eso: en vez de fundar todo en los mismos conceptos clásicos (como posición, velocidad, ...), va ha hacer una análisis crítico de esos conceptos, introduciendo nuevas formas de entenderlos.

It is well known that the formal rules which are used in quantum theory for calculating observable quantities such as the energy of the hydrogen atom may be seriously criticized on the grounds that they contain, as basic element, relationships between quantities that are apparently unobservable in principle, e.g., position and period of revolution of the electron. Thus these rules lack an evident physical foundation, unless one still wants to retain the hope that the hitherto unobservable quantities may later come within the realm of  experimental determination. This hope might be regarded as justified if the above-mentioned rules were internally consistent and applicable to a clearly defined range of quantum mechanical problems. Experience however shows that only the hydrogen atom and its Stark effect are amenable to treatment by these formal rules of quantum theory. Fundamental difficulties already arise in the problem of ' crossed fields' (hydrogen atom in electric and magnetic fields of differing directions). Also, the reaction of atoms to periodically varying fields cannot be described by these rules. Finally, the extension of the quantum rules to the treatment of atoms having several electrons has proved unfeasible.

El efecto Stark, junto con el efecto Zeeman fueron grandes problemas que tuvieron que ser resueltos en aquella época. El primero involucra un campo eléctrico, el segundo es provocado por un campo magnético. Explicar los resultados de los espectros de los átomos sometidos a tales campos era un desafío para la física.

It has become the practice to characterize this failure of the  quantum-theoretical rules as a deviation from classical mechanics, since the rules themselves were essentially derived from classical mechanics. This characterization has, however, little meaning when one realizes that the Einstein-Bohr frequency condition (which is valid in all cases) already represents such a complete departure from classical mechanics, or rather (using the viewpoint of wave theory) from the kinematics underlying this mechanics, that even for the simplest quantum-theoretical problems the validity of classical mechanics simply cannot be maintained. In this situation it seems sensible to discard all hope of observing hitherto unobservable quantities, such as the position and period of the electron, and to concede that the partial agreement of the quantum rules with experience is more or less fortuitous. Instead it seems more reasonable to try to establish a theoretical quantum mechanics, analogous to classical mechanics, but in which only  relations between observable quantities occur. One can regard the frequency condition and the dispersion theory of Kramers together
with its extensions in recent papers as the most important first steps toward such a quantum-theoretical mechanics. In this paper, we shall seek to establish some new quantum-mechanical relations and apply these to the detailed treatment of a few special problems. We shall restrict ourselves to problems involving one degree of freedom.

Es decir, se va a limitar a una sola "coordenada". Vemos que menciona a Kramer. Tenemos que ver cuáles eran esas ideas de teoría de la dispersión, porque algunas de esas ideas y fórmulas terminan apareciendo en este "paper" de Heisenberg. El conocía a Kramer, por ser el ayudante principal de Bohr, y trabajó con él y publicó "papers" conjuntos cuando estuvo de visita en Copenhague.

Y recuerda las condiciones de frecuencia de Einstein-Bohr, una gran sorpresa que relaciona energía con frecuencia, y que había sido reanimada con las ideas de de Broglie publicadas unos meses antes de este "paper" (sin embargo, no parece que lo de de Broglie influyera en el desarrollo de las ideas de Heinsenberg).

Entonces, estaba todo dado para explicar las frecuencias espectrales, en gran parte al trabajo fundacional de Bohr, pero no se había podido avanzar tanto en el cálculo de las INTENSIDADES de esas frecuencias, por ejemplo, cuando un tipo de átomo era sometido a radiación y temperatura.

Hay un buen resumen de lo que tenemos que entender de este "paper" en:

Papers from the beginning of quantum mechanics

Leo ahí:

Heisenberg's original matrix mechanics - This is the work that created the modern theory of quantum mechanics (Heisenberg 1925). Heisenberg wanted to tackle the question of how to predict correctly the intensities of atomic transition lines, as Bohr had already clarified how to obtain the transition frequencies. Heisenberg began by noticing that, according to Bohr, the correct quantum transition frequencies do not depend just on the current state of motion (as do the frequencies of emitted radiation for a classical orbit), but rather on two states (initial and final). Likewise, in classical theory, the intensities of emitted radiation would be given by the squares of the Fourier amplitudes of the oscillating dipole moment for a given orbit. In an ingenious step, Heisenberg then postulated that instead of a set of Fourier amplitudes for a given orbit (enumerated by one index), one would have to introduce a set of amplitudes depending on two indices, one for the initial, the other for the final state. He assumed that the equations of motion for those amplitudes looked formally the same as in classical theory (Heisenberg equations of motion). The last crucial ingredient is the commutation relation. This he derived by looking at the linear response of an electron to an external perturbation (essentially deriving something like Kubo's formula, containing the commutator) and then demanding that the short-time response would be always that of a free, classical electron. This fixes the commutator between position and momentum. Thus was born matrix mechanics. He applied this immediately to the harmonic oscillator and also dealt with the anharmonic oscillator using perturbation theory.

See also Heisenberg's Nobel Lecture from 1933 to learn more about his view on these developments, and the slightly earlier overview (Heisenberg 1928 (Naturwissenschaften)) that also includes much of the developments before matrix mechanics.

Vamos a tener que visitar series de Fourier, y conceptos clásicos para entender "la maiga" que termina haciendo Heisenberg. Agregaría a lo de arriba que hay unas conferencias de Heinserberg en América (1930?), que tienen una mejor explicación de sus ideas, con más detalle y gentileza que en este "paper", donde a veces da algunos saltos que nos evidentes.

Nos leemos!

Angel "Java" Lopez
http://www.ajlopez.com
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Por ajlopez, en: Ciencia